pascal's triangle Archives - Page 2 of 2

Milkshakes and Power Sets

October 3, 2011 GB College Mathematics, Combinatorics, Set Theory

In Milkshakes, Beads, and Pascal’s Triangles, we have talked about a systematic way of choosing a combination of objects from a larger number of objects. Let us recall the problem in the said post.

Issa went to a shake kiosk and want to buy a milkshake. The shake vendor told her that she can choose plain milk, or she can choose to combine any number of flavors in any way she wants. There are four flavors to choose from: Apple, Banana, Chico, and Durian. How many possible combination of flavors can Issa make?

In the problem, Issa can choose any number of flavors and any combination. She can choose plain milk, choose one flavor at a time, two flavors at a time, three flavors at a time, or four flavors at a time as shown in the table below (click the table to enlarge).

Notice that in writing the list, we have exhausted the number of subsets in a set with four elements. If we let a, b, c, and d stand for avocado, banana, chico, and durian, let them be members or a set and use the set notation, we can write the subsets as follows: » Read more

One comment binomial coefficient, binomial coefficient sum, combination, pascal's triangle, power sets, sets, subsets

The Binomial Expansion

July 5, 2010 GB College Mathematics, Combinatorics, High School Mathematics

Note: This is the second part of the Binomial Expansion Series

Part I: Milkshakes, Beads, and Pascal’s Triangle

Part II: Binomial Expansion

In the Milkshakes, Beads, and Pascal’s Triangle article, we have shown that the combination of the binary numbers 1 and 0 may be interpreted as the number the flavors of milk shakes, or the number of possible paths of the bead in our Galton board as shown in Table 1. Recall that in the Milkshake problem, Issa was given a choice to combine any number of flavors from four fruits: Apple, Banana, Chico and Durian. Thus, 0101 means banana-durian milkshake. On the other hand, in the beads problem, 0101 is LRLR or the bead went to the left after hitting the peg in row A, right in after hitting the peg in row B, left after hitting the peg in row C and right after hitting the peg in row D. » Read more

One comment binomial expansion, binomial expansion help, binomial theorem, Galton board, n objects taken k at a time, pascal's triangle, permutations and combinations

Milkshakes, Beads, and Pascal’s Triangle

June 28, 2010 GB College Mathematics, Combinatorics, High School Mathematics

Note: This is the first part of the Binomial Theorem Series

Binomial Theorem I: Milkshakes, Beads, and Pascal’s Triangle

Binomial Theorem II: The Binomial Expansion

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The Milk Shake Problem

Problem 1: Issa went to a shake kiosk and want to buy a milkshake. The shake vendor told her that she can choose plain milk, or she can choose to combine any number of flavors in any way she want. There are four flavors to choose from: Apple, Banana, Chico, and Durian.

How many possible combinations can Issa create?

It is clear that Issa can choose her milk shake to have no flavor (pure milk shake), one flavor, two flavors, three flavors, or four flavors. The possible combinations are shown below.

Table 1

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